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Aim Species–area relationships (SARs) are fundamental scaling laws in ecology although their shape is still disputed. At larger areas, power laws best represent SARs. Yet, it remains unclear whether SARs follow other shapes at finer spatial grains in continuous vegetation. We asked which function describes SARs best at small grains and explored how sampling methodology or the environment influence SAR shape. Location Palaearctic grasslands and other non‐forested habitats. Taxa Vascular plants, bryophytes and lichens. Methods We used the GrassPlot database, containing standardized vegetation‐plot data from vascular plants, bryophytes and lichens spanning a wide range of grassland types throughout the Palaearctic and including 2,057 nested‐plot series with at least seven grain sizes ranging from 1 cm2 to 1,024 m2. Using nonlinear regression, we assessed the appropriateness of different SAR functions (power, power quadratic, power breakpoint, logarithmic, Michaelis–Menten). Based on AICc, we tested whether the ranking of functions differed among taxonomic groups, methodological settings, biomes or vegetation types. Results The power function was the most suitable function across the studied taxonomic groups. The superiority of this function increased from lichens to bryophytes to vascular plants to all three taxonomic groups together. The sampling method was highly influential as rooted presence sampling decreased the performance of the power function. By contrast, biome and vegetation type had practically no influence on the superiority of the power law. Main conclusions We conclude that SARs of sessile organisms at smaller spatial grains are best approximated by a power function. This coincides with several other comprehensive studies of SARs at different grain sizes and for different taxa, thus supporting the general appropriateness of the power function for modelling species diversity over a wide range of grain sizes. The poor performance of the Michaelis–Menten function demonstrates that richness within plant communities generally does not approach any saturation, thus calling into question the concept of minimal area.

Bayesian inference methods rely on numerical algorithms for both model selection and parameter inference. In general, these algorithms require a high computational effort to yield reliable estimates. One of the major challenges in phylogenetics is the estimation of the marginal likelihood. This quantity is commonly used for comparing different evolutionary models, but its calculation, even for simple models, incurs high computational cost. Another interesting challenge relates to the estimation of the posterior distribution. Often, long Markov chains are required to get sufficient samples to carry out parameter inference, especially for tree distributions. In general, these problems are addressed separately by using different procedures. Nested sampling (NS) is a Bayesian computation algorithm, which provides the means to estimate marginal likelihoods together with their uncertainties, and to sample fromthe posterior distribution at no extra cost. The methods currently used in phylogenetics for marginal likelihood estimation lack in practicality due to their dependence on many tuning parameters and their inability of most implementations to provide a direct way to calculate the uncertainties associated with the estimates, unlike NS. In this article, we introduce NS to phylogenetics. Its performance is analysed under different scenarios and compared to established methods. We conclude that NS is a competitive and attractive algorithm for phylogenetic inference. An implementation is available as a package for BEAST 2 under the LGPL licence, accessible at https://github.com/BEAST2-Dev/nested-sampling.

We present an improved version of the nested sampling algorithm nessai in which the core algorithm is modified to use importance weights. In the modified algorithm, samples are drawn from a mixture of normalising flows and the requirement for samples to be independently and identically distributed (i.i.d.) according to the prior is relaxed. Furthermore, it allows for samples to be added in any order, independently of a likelihood constraint, and for the evidence to be updated with batches of samples. We call the modified algorithm i-nessai . We first validate i-nessai using analytic likelihoods with known Bayesian evidences and show that the evidence estimates are unbiased in up to 32 dimensions. We compare i-nessai to standard nessai for the analytic likelihoods and the Rosenbrock likelihood, the results show that i-nessai is consistent with nessai whilst producing more precise evidence estimates. We then test i-nessai on 64 simulated gravitational-wave signals from binary black hole coalescence and show that it produces unbiased estimates of the parameters. We compare our results to those obtained using standard nessai and dynesty and find that i-nessai requires 2.68 and 13.3 times fewer likelihood evaluations to converge, respectively. We also test i-nessai of an 80 s simulated binary neutron star signal using a reduced-order-quadrature basis and find that, on average, it converges in 24 min, whilst only requiring 1.01 × 10 6 likelihood evaluations compared to 1.42 × 10 6 for nessai and 4.30 × 10 7 for dynesty . These results demonstrate that i-nessai is consistent with nessai and dynesty whilst also being more efficient.

Multi-fidelity (MF) surrogate models have shown great potential in simulation-based design since they can make a trade-off between high prediction accuracy and low computational cost by augmenting the small number of expensive high-fidelity (HF) samples with a large number of cheap low-fidelity (LF) data. In this work, a generalized hierarchical co-Kriging (GCK) surrogate model is proposed for MF data fusion with both nested and non-nested sampling data. Specifically, a comprehensive Gaussian process (GP) Bayesian framework is developed by aggregating calibrated LF Kriging model and discrepancy stochastic Kriging model. The stochastic Kriging model enables the GCK model to consider the predictive uncertainty from the LF Kriging model at HF sampling points, making it possible to estimate the model parameter separately under both nested and non-nested sampling data. The performance of the GCK model is compared with three well-known Kriging-based MF surrogates, i.e., hybrid Kriging–scaling (HKS) model, KOH autoregressive (KOH) model, and hierarchical Kriging (HK) model, by testing them on two numerical examples and two real-life cases. The influence of correlations between LF and HF samples and the cost ratio between them are also analyzed. Comparison results on the illustrated cases demonstrate that the proposed GCK model shows great potential in MF modeling under non-nested sampling data, especially when the correlations between LF and HF samples are weak.

As an effective means of structural vibration suppression, piezoelectric actuator vibration suppression system(PAVSS) has been gradually applied in structural vibration control. However, the performance degradation or failure of piezoelectric actuator under long service life will lead to the degradation of service reliability of PAVSS, so the scheme obtained by deterministic optimization method is difficult to ensure the service reliability. Aiming at the reliability evaluation and optimal design of PAVSS in service, the failure mechanism considering of the performance degradation or failure of piezoelectric actuator is analyzed, and the reliability evaluation method based on nested sampling and weighted statistics is constructed, which solves the reliability evaluation problem of PAVSS with the characteristics of load-sharing and redundancy. A reliability-based optimization method based on master-slave parallel genetic algorithm is proposed to optimize the position and angle of the actuators. At last, the feasibility and effectiveness of the proposed method are proved by an example, which not only provides a theoretical basis for the high reliability design of PAVSS in service, but also provides a new solution for the engineering problem of structural vibration control. 作为结构振动抑制的有效手段, 压电作动器振动抑制系统在结构振动控制中逐步得到应用。但是长服役期内压电作动器的性能退化/失效会导致整个振动抑制系统的服役可靠性退化, 确定性优化方法得到的方案难以保证系统的服役可靠性。针对服役期内压电振动抑制系统的可靠性评估及优化设计问题, 分析了压电作动器性能退化和失效下振动抑制系统的失效机理, 构建了基于嵌套抽样和加权统计的振动抑制系统可靠性评估方法, 解决了具有载荷共享和冗余特征的振动抑制系统可靠性评估问题, 提出了基于主从式并行遗传算法的压电作动器可靠性布局优化方法, 实现了作动器的位置和角度优化。通过算例证明了可靠性布局优化的可行性和有效性, 为服役期内振动抑制系统的高可靠性设计提供理论基础, 也可为结构振动控制这一工程问题提供新的解决思路。

Nested sampling is an efficient method for calculating Bayesian evidence in data analysis and partition functions of potential energies. It is based on an exploration using a dynamical set of sampling points that evolves to higher values of the sampled function. When several maxima are present, this exploration can be a very difficult task. Different codes implement different strategies. Local maxima are generally treated separately, applying cluster recognition of the sampling points based on machine learning methods. We present here the development and implementation of different search and clustering methods on the nested_fit code. Slice sampling and the uniform search method are added in addition to the random walk already implemented. Three new cluster recognition methods are also developed. The efficiency of the different strategies, in terms of accuracy and number of likelihood calls, is compared considering a series of benchmark tests, including model comparison and a harmonic energy potential. Slice sampling proves to be the most stable and accurate search strategy. The different clustering methods present similar results but with very different computing time and scaling. Different choices of the stopping criterion of the algorithm, another critical issue of nested sampling, are also investigated with the harmonic energy potential.

Nested sampling (NS) is a stochastic method for computing the log-evidence of a Bayesian problem. It relies on stochastic estimates of prior volumes enclosed by likelihood contours, which limits the accuracy of the log-evidence calculation. We propose to transform the prior volume estimation into a Bayesian inference problem, which allows us to incorporate a smoothness assumption for likelihood–prior–volume relations. As a result, we aim to increase the accuracy of the volume estimates and thus improve the overall log-evidence calculation using NS. The method presented works as a post-processing step for NS and provides posterior samples of the likelihood–prior–volume relation, from which the log-evidence can be calculated. We demonstrate an implementation of the algorithm and compare its results with plain NS on two synthetic datasets for which the underlying evidence is known. We find a significant improvement in accuracy for runs with less than one hundred active samples in NS but a proneness for numerical problems beyond this point.

The epistemic uncertainty in coronavirus disease (COVID-19) model-based predictions using complex noisy data greatly affects the accuracy of pandemic trend and state estimations. Quantifying the uncertainty of COVID-19 trends caused by different unobserved hidden variables is needed to evaluate the accuracy of the predictions for complex compartmental epidemiological models. A new approach for estimating the measurement noise covariance from real COVID-19 pandemic data has been presented based on the marginal likelihood (Bayesian evidence) for Bayesian model selection of the stochastic part of the Extended Kalman filter (EKF), with a sixth-order nonlinear epidemic model, known as the SEIQRD (Susceptible–Exposed–Infected–Quarantined–Recovered–Dead) compartmental model. This study presents a method for testing the noise covariance in cases of dependence or independence between the infected and death errors, to better understand their impact on the predictive accuracy and reliability of EKF statistical models. The proposed approach is able to reduce the error in the quantity of interest compared to the arbitrarily chosen values in the EKF estimation.

Nested sampling is a simulation method for approximating marginal likelihoods. We establish that nested sampling has an approximation error that vanishes at the standard Monte Carlo rate and that this error is asymptotically Gaussian. It is shown that the asymptotic variance of the nested sampling approximation typically grows linearly with the dimension of the parameter. We discuss the applicability and efficiency of nested sampling in realistic problems, and compare it with two current methods for computing marginal likelihood. Finally, we propose an extension that avoids resorting to Markov chain Monte Carlo simulation to obtain the simulated points.